Stiness-oriented numerical model for non-linear reinforced concrete beam systems Master Thesis presentation SEng Agnieszka KNOPPIKWRÓBEL Supervised by: PhD SEng Grzegorz WANDZIK Silesian University of Technology Faculty of Civil Engineering Gliwice, 9 Feb 2011
Presentation outline Introduction 1 Introduction Objective of thesis Range of thesis 2 Structural analysis Problems in standards 3 Static scheme Material model Cross-section model Bending stiness Static analysis with FEM 4 5 Objective of thesis Range of thesis
Objective of thesis Range of thesis Objective of thesis To derive a numerical model for designing of exural reinforced concrete beams taking into consideration non-linear behaviour of reinforced concrete and redistribution of internal forces as a result of stiness degradation of the elements due to crack formation at exure, thus providing a unied algorithm for static calculations and dimensioning.
Objective of thesis Range of thesis Range of thesis 1 methods of non-linear analysis 2 theory of stiness degradation Practical solution 1 numerical model
Structural analysis Problems in standards Linear-elastic analysis The most simplied method, but with inconsistencies: Ultimate Limit State linear analysis in static calculations M Sd plastic properties considered in cross-section resistance M Rd concrete section only Serviceability Limit State modied stiness values B I and B II cracking moment M cr creep and shrinkage (in parametric form)
Structural analysis Problems in standards Linear-elastic analysis with limited redistribution Bending moments distribution resulting from linear-elastic analysis redistributed: δ = M red M el 100% Yielding of steel ensured by a specic value of reinforcement strain or a depth of neutral axis limitation. Resulting allowable redistribution: European standards δ 30% American standards δ 20%
Structural analysis Problems in standards Plastic analysis Introduction to plastic properties of concrete: plastic hinge, rotational capacity, ductility conditions. Method taking into account plastic properties of concrete for beam elements: limit equilibrium method.
Structural analysis Problems in standards Non-linear analysis Applicable for both ULS and SLS, provided that equilibrium and compatibility are satised and an adequate non-linear behaviour for materials is assumed. No consistent design procedure is provided except for some design requirements which must be satised.
Structural analysis Problems in standards Problems with non-linear analysis in standards little or no information; poor and extensively simplied representation of material properties, applicability to a very narrow range of engineering problems, need of reference to literature with no available consistent solution, only numerical analysis possible, great designer experience needed.
Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Geometry Arbitrary statically-indeterminate beam with known geometry, reinforcement, material properties and loading pattern. Figure 1: Exemplary continuous beam
Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Discretisation Division into segments of the same cross-section type. Assignment of initial stiness. Figure 2: Discretisation of the beam
Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Models for concrete and steel CONCRETE: MC2010 model for compression and tension: STEEL: EC2 model, steel as isotropic material: Figure 3: Material model for concrete Figure 4: Material model for steel
Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Curvature Curvatures κ k taken at k equal intervals for α k ( 90, 90 ): Boundary values for x k = 0 and x k = h: (a) κ > 0 (b) κ < 0 Figure 5: Geometrical interpretation of curvature Figure 6: N 0k and N hk
Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Bending moment Axial force determined in each sub-step for x kf until for x kf N k = 0. Then values of bending moments M k in pure bending in each k step for κ k calculated. (a) cross-section (b) strains (c) stresses (d) internal forces Figure 7: Determination of internal forces in section with numerical integration
Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Momentcurvature relationship Momentcurvature relationship derived for each cross-section type j in both positive and negative range of bending moments (curvatures). Figure 8: Hypothetical Mκ diagram
Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Stiness of cross-section Stiness must be determined in every cross-section after each r th loading step. Figure 9: Geometrical interpretation of section stiness Methods to determine the value of stiness: 1 nite dierence methods 2 curve tting interpolation approximation spline
Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Stiness of segment Stiness of each segment i of a given cross-section type j = t(i) is taken as a mean value calculated for the maximum bending moment M i,r acting in that segment after r th loading step, taking into account the tension stiening eect coecient ξ i,r and stiness of a single cross-section B t(i),r : B mean,t(i),r = (1 ξ i,r )B t(i),r Stiness is changed within a segment when cracking occurs. Stiness cannot be regained, cracking cannot be reversed
Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Static calculations General assumptions displacement nal element method used incremental method used for small increments to avoid necessity of iterations linearisation of K(u)u = P problem into Ku = P Calculations constant increments of load q bending moments determined in each node the maximum value of bending moment taken for the segment modication of stiness after each loading step
Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Static calculations Figure 10: Step 1. Initial loading
Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Static calculations Figure 11: Step R 1. Formation of the rst plastic hinge
Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Static calculations Figure 12: Step R 2. Formation of the last plastic hinge
Static scheme Material model Cross-section model Bending stiness Static analysis with FEM Bending moment distribution 1 determination of stiness matrix and vector of nodal loads of each segment 2 formation of global stiness matrix and global vector of nodal loads (aggregation) global stiness matrix is a band matrix only entries on diagonal are aggregated 3 denition of vector of nodal displacements 4 introduction of boundary (support) conditions 5 solution u = K 1 P equation 6 calculation M and V forces
General conclusions design process requires idealisations model proposed approaches too simplied, inconsistent, with little information provided non-linear analysis must be used but it requires experience and numerical calculations numerical simulation requires development of numerical model and co-operation between scientist/engineer and software developer
Summary and future prospects objective of this thesis was achieved procedure was described in a very detail mathematical and geometrical interpretations were provided graphical representations were shown model is ready for implementation by a software expert not necessarily familiar with the discussed physical problem validation and evaluation of redistribution as a result of stiness degradation will be possible
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